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Circuits, Systems, and Signal Processing

, Volume 38, Issue 5, pp 1962–1981 | Cite as

Stabilization of Positive 2D Fractional-Order Continuous-Time Systems with Delays

  • Laila Dami
  • Mohamed Benhayoun
  • Abdellah BenzaouiaEmail author
Article
  • 31 Downloads

Abstract

This paper is concerned with the stability and the stabilization problems for a class of 2D fractional-order positive linear systems with and without delays. The obtained results are based on Lyapunov–Krasovskii function. Asymptotic stability and stabilization criteria are then derived. The synthesis of the state controller is obtained by giving conditions in terms of linear programs. A discretization method is established for the 2D fractional-order continuous-time system with delays in order to facilitate the simulation of the acquired results. The accuracy, efficiency and convergence of the obtained results are shown through numerical examples.

Keywords

2D positive system Fractional-order system Stabilization Delayed systems Discretization Darboux equation 

References

  1. 1.
    A. Atangana, On the stability and convergence of the time-fractional variable order telegraph equation. J. Comput. Phys. 293, 104–114 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    A. Benzaouia, A. Hmamed, F. Tadeo, Two-Dimensional Systems from Introduction to the State of Art (Springer, Berlin, 2016)CrossRefzbMATHGoogle Scholar
  3. 3.
    A. Benzaouia, F. Mesquine, M. Benhayoun, Stabilization of continuous-time fractional positive systems with delays, in 5th international conference on systems and control (ICSC), (2016)Google Scholar
  4. 4.
    A. Benzaouia, F. Mesquine, M. Benhayoun, Stabilization of continuous-time fractional positive systems by using a Lyapunov function. IEEE Trans. Autom. Control 59, 2203–2208 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    A. Benzaouia, A. Hmamed, F. Tadeo, Continuous-time fractional bounded positive systems, in 52nd IEEE conference on decision and control, (2013)Google Scholar
  6. 6.
    A. Benzaouia, M. Benhayoun, State-feedback stabilization of 2-D continuous systems with delays. Int. J. Innov. Comput. Inf. Control 7, 977–988 (2011)Google Scholar
  7. 7.
    M. Benhayoun, F.B. Mesquine, A. Benzaouia, Delay-dependent stabilizability of 2-D delayed continuous systems with saturating control. Circuits Syst. Signal Process 32, 2723–2743 (2013)MathSciNetCrossRefGoogle Scholar
  8. 8.
    M. Buslowicz, Stability of linear continuous-time fractional order systems with delays of the retarded type. Bull. Pol. Acad. Tech. 56, 319–324 (2008)Google Scholar
  9. 9.
    L. Debnath, Recent applications of fractional calculus to science and engineering. Int. J. Math. Math. Sci. 2003, 3413–3442 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    J. Devi, F. Mc, Z. Rae, Drici variational Lyapunov method for fractional differential equations. Comput. Math. Appl. 64, 2982–2989 (2012)MathSciNetCrossRefGoogle Scholar
  11. 11.
    C. Du, L. Xie, H1 Control and Filtering of Two-Dimensional Systems (Springer, Berlin, 2002)Google Scholar
  12. 12.
    E. Fornasini, G. Marchesini, Doubly indexed dynamical systems: state-space models and structural properties. IEEE Trans. Autom. Control 12, 59–72 (1978)MathSciNetzbMATHGoogle Scholar
  13. 13.
    E. Fornasini, G. Marchesini, State-space realization theory of two dimensional filters. IEEE Trans. Autom. Control 21, 484–492 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    G.H. Gao, Z.Z. Sun, A compact difference scheme for the fractional subdiffusion equations. J. Comput. Phys. 230, 586–595 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    J.F. Gomez-Aguilara, M. Miranda-Hernandez, M.G. Lopez-Lopez, V.M. Alvarado-Martinez, D. Baleanud, Modeling and simulation of the fractional space-time diffusion equation. Commun. Nonlinear Sci. Numer. Simul. 30, 115–127 (2016)MathSciNetCrossRefGoogle Scholar
  16. 16.
    A. Hmamed, F. Mesquine, M. Benhayoun, A. Benzaouia, F. Tadeo, Stabilization of 2-D saturated systems by state feedback control. Multidimens. Syst. Signal Process 21, 277–292 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    A. Hmamed, M. Alfidi, A. Benzaouia, F. Tadeo, LMI conditions for robust stability of 2D linear discrete-time systems, Math. Probl. Eng (2008). https://doi.org/10.1155/2008/356124
  18. 18.
    A. Hmamed, M. Ait Rami, M. Alfidi, Controller synthesis for positive 2D systems described by the Roesser model, in 47th IEEE conference on decision and control, (2008)Google Scholar
  19. 19.
    S. Huang, Z. Xiang, Delay-dependent robust \(\text{ H }\infty \) control for 2-D discrete nonlinear systems with state delays. Multidimens. Syst. Signal Process 25, 775–794 (2014)CrossRefzbMATHGoogle Scholar
  20. 20.
    H. Jian-Bing, H. Yan, Z. Ling-Dong, A novel stability theorem for fractional systems and its applying in synchronizing fractional chaotic system based on back-stepping approach. Acta Phys. Sin. 58, 2235–2239 (2009)zbMATHGoogle Scholar
  21. 21.
    T. Kczorek, Positive switched 2D linear systems described by the Roesser models. Eur. J. Control 18, 239–246 (2012)MathSciNetCrossRefGoogle Scholar
  22. 22.
    T. Kaczorek, Selected Problems of Fractional Systems Theory (Springer, Berlin, 2011), p. 411CrossRefzbMATHGoogle Scholar
  23. 23.
    T. Kaczorek, Positive different orders fractional 2D linear systems. Bull. Pol. Acad. Sci. 58, 453–458 (2010)zbMATHGoogle Scholar
  24. 24.
    T. Kaczorek, Positivity and stabilization of fractional 2D linear systems described by the Roesser model. IFAC Proc. Vol. 42, 256–261 (2009)CrossRefGoogle Scholar
  25. 25.
    T. Kaczorek, Positive 1D and 2D Systems (Springer, Berlin, 2001)zbMATHGoogle Scholar
  26. 26.
    T. Kaczorek, Two-Dimensional Linear Systems (Springer, Berlin, 1985)zbMATHGoogle Scholar
  27. 27.
    T.A.M. Langlands, B.I. Henry, The accuracy and stability of an implicit solution method for the fractional diffusion equation. J. Comput. Phys. 205, 719–736 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    W.S. Lu, A. Antoniou, Two-Dimensional Digital Filters (Marcel Dekker, New York, 1992)zbMATHGoogle Scholar
  29. 29.
    A. Oustaloup, B. Mathieu, La Commande Crone (Editions Hermes, Paris, 1999)Google Scholar
  30. 30.
    P. Przyborowski, T. Kaczorek, Positive 2D discrete-time linear Lyapunov systems. Int. J. Appl. Math. Comput. Sci 19, 95105 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    R.P. Roesser, A discrete state-space model for linear image processing. IEEE Trans. Autom. Control 20, 1–10 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    J. Sabatier, M. Moze, C. Farges, LMI stability conditions for fractional order systems. Comput. Math. Appl. 59, 1594–1609 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    E. Sadat, A. Shahri, S. Balochian, An analysis and design method for fractional-order linear systems subject to actuator saturation and disturbance. Optim. Control Appl. Methods 37, 305–322 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    E. Sadat, A. Shahri, S. Balochian, Analysis of fractional-order linear systems with saturation using Lyapunov’s second method and convex optimization. Int. J. Autom. Comput. 12, 440–447 (2015)CrossRefGoogle Scholar
  35. 35.
    V. Singh, New approach to stability of 2-D discrete systems with state saturation. Signal Process 92, 240–247 (2012)CrossRefGoogle Scholar
  36. 36.
    S.B. Yuste, Weighted average finite difference methods for fractional diffusion equations. J. Comput. Phys. 216, 264274 (2006)MathSciNetCrossRefGoogle Scholar
  37. 37.
    P. Zhuang, H.L. Liao, Z.Z. Sun, Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems. Commun. Nonlinear Sci. Numer. Simul. 22, 650–659 (2015)MathSciNetCrossRefGoogle Scholar
  38. 38.
    P. Zhuang, F. Liu, A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications. J. Comput. Phys. 259, 3350 (2014)MathSciNetGoogle Scholar
  39. 39.
    P. Zhuang, H.L. Liao, Z.Z. Sun, Finite difference methods for the time fractional diffusion equation on non-uniform meshes. J. Comput. Phys. 265, 195–210 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.LAEPT, Department of PhysicsUniversity Cadi AyyadMarrakeshMorocco

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